Chapter 4: Measuring motion

4.2 Velocity

Velocity is the rate of change of displacement: velocity tells us how an object’s position is changing in time.

\[
\text{velocity} = \frac{\text{change in position}}{\text{change in time}}
\]

Or, in symbols (for motion along only the \(x\) axis):

\[v_x = \frac{\Delta x}{\Delta t} \tag{4.1}\]

The symbol \(\Delta\) is a capital Greek letter delta, and it always refers to a change. Mathematically, delta refers to the difference between final and initial value of a particular quantity. For example \(\Delta x\) is the change in the \(x\) coordinate of an object’s position, and is given by

\[
\Delta x = x_f – x_i \tag{4.2}
\]

Note how we use subscripts to be a bit more specific in what we’re referring to with each variable. In the equation defining \(v\) we used the subscript \(x\) to indicate velocity along the \(x\) axis, and in equation defining \(\Delta\) we used subscripts \(f\) and \(i\) to indicate the final and initial values of the variable \(x\).

It is common practice to say \(t_i = 0\) and \(t_f = t\). This means we often simply use \(t\) instead of \(\Delta t\). This reflects the practice that \(t\) usually refers to some elapsed time, such as you would measure with a stopwatch.

Because position is a vector, velocity is also a vector. The direction of the velocity vector is the direction of motion. When working with one-dimensional motion, a positive or negative sign indicates direction; for two-dimensional motion we’ll use vector notation. For two-dimensional motion, we use the rate of change of each component of the position:

\[
\vec{v} = \underbrace{\left(\frac{\Delta x}{\Delta t}\right)}_{v_x}\hat{x} + \underbrace{\left(\frac{\Delta y}{\Delta t}\right)}_{v_y}\hat{y} \tag{4.3}
\]

The magnitude of an object’s velocity is its speed:

\[
v = \sqrt{\left(v_x\right)^2 + \left(v_y\right)^2} = \text{speed}
\]

(Recall from chapter 2 that the symbol \(v\) represents the magnitude of vector \(\vec{v}\).)

The SI unit for velocity (and speed) is meters per second (m/s).

Example

A cat is asleep, when a bad dream suddenly wakes it up and it runs across the room. The cat wakes up at the time 1:42:03, and reaches the other side of the room at 1:42:11. The position of the box the cat was sleeping in is

\[
\vec{r}_1 = x_1\hat{x} + y_1\hat{y} = 12\hat{x} – 4\hat{y}\ \textrm{m}
\]

and the position the cat ends at is

\[
\vec{r}_2 = x_2\hat{x} + y_2\hat{y} = 2\hat{x} + 8\hat{y}\ \textrm{m}
\]

What were the cat’s velocity and speed?

Velocity is the rate of change of position. With 2D motion, we need to consider both vector components.

\[
\begin{align*}
\vec{v} &= \frac{\Delta \vec{r}}{\Delta t} \\
&= \frac{\Delta x}{\Delta t}\hat{x} + \frac{\Delta y}{\Delta t}\hat{y} \\
&= \frac{x_2 – x_1}{\Delta t}\hat{x} + \frac{y_2 – y_1}{\Delta t}\hat{y} \\
&= \frac{2 – 12\ \textrm{m}}{8\ \textrm{s}}\hat{x} + \frac{8 – (-4)\ \textrm{m}}{8\ \textrm{s}}\hat{y} \\
&= -1.25\hat{x} + 1.5\hat{y}\ \textrm{m/s}
\end{align*}
\]

Note that the cat’s velocity is a vector, given here in component form. The cat’s speed is the magnitude of its velocity:

\[
\begin{align*}
v &= \sqrt{v_x^2 + v_y^2} \\
&= \sqrt{(-1.25\ \textrm{m/s})^2 + (1.5\ \textrm{m/s})^2} \\
&= 1.95\ \textrm{m/s}
\end{align*}
\]

Practice

Practice

Practice

Practice